# Block design

**Editor-In-Chief:** C. Michael Gibson, M.S., M.D. [1]

## Overview

In combinatorial mathematics, a **block design** (more fully, a **balanced incomplete block design**) is a particular kind of set system, which has long-standing applications to experimental design (an area of statistics) as well as purely combinatorial aspects.

Given a finite set *X* (of elements called points) and integers *k*, *r*, λ ≥ 1, we define a **2-design** *B* to be a set of *k*-element subsets of *X*, called **blocks**, such that the number *r* of blocks containing *x* in *X* is independent of *x*, and the number λ of blocks containing given distinct points *x* and *y* in *X* is also independent of the choices.

Here *v* (the number of elements of *X*, called points), *b* (the number of blocks), *k*, *r*, and λ are the **parameters** of the design. (Also, *B* may not consist of all *k*-element subsets of *X*; that is the meaning of *incomplete*.) The design is called a **( v, k, λ)-design** or a

**(**. The parameters are not all independent;

*v*,*b*,*r*,*k*, λ)-design*v*,

*k*, and λ determine

*b*and

*r*, and not all combinations of

*v*,

*k*, and λ are possible. The two basic equations connecting these parameters are

A fundamental theorem (**Fisher's inequality**) is that *b* ≥ *v* in any block design. The case of equality is called a symmetric design; it has many special features.

Examples of block designs include the lines in finite projective planes (where *X* is the set of points of the plane and λ = 1), and Steiner triple systems (*k* = 3). The former is a relatively simple example of a symmetric design.

## Generalization: *t*-designs

Given any integer *t* ≥ 2, a *t*-design*B* is a class of *k*-element subsets of *X* (the set of points) , called **blocks**, such that the number *r* of blocks that contain any point *x* in *X* is independent of *x*, and the number λ of blocks that contain any given *t*-element subset *T* is independent of the choice of *T*. The numbers *v* (the number of elements of *X*), *b* (the number of blocks), *k*, *r*, λ, and *t* are the **parameters** of the design. The design may be called a ** t-(v,k,λ)-design**. Again, these four numbers determine

*b*and

*r*and the four numbers themselves cannot be chosen arbitrarily. The equations are

where *b _{i}* is the number of blocks that contain any

*i*-element set of points.

Examples include the *d*-dimensional subspaces of a finite projective geometry (where *t* = *d* + 1 and λ = 1).

The term *block design* by itself usually means a 2-design.

## References

- van Lint, J.H., and R.M. Wilson (1992),
*A Course in Combinatorics*. Cambridge, Eng.: Cambridge University Press. - Template:Mathworld